I am trying to grasp the difference between IV-estimation and Heckman's selection model.
I do that by considering the following set-up. My outcome if interest is
y (I observe it for all observations, incidental truncation is not the problem here!). I have an endogenous binary treatment variable
w. Also, I have a binary control function
x. Finally, I have an instrument
z for my binary treatment variable
w. I know how to use instrumental variables, but I have a hard time understanding Heckman's selection model in this case. The first stage should be a probit model, where I regress
z. But I don't understand how my second stage is defined and which assumptions it relies on. I would like to know how to estimate both the average treatment effect and the average treatment effect on the treated. And also how the exclusion restriction kicks in (including or not including
z in the second stage).
Hope that someone can provide intuition.